The Physics of Granular Materials

1. Application of Statistics and Percolation Theory The Physics of Granular Materials Temmy Brotherson Michael Lam

2. Granular Materials • What are granular materials? • Macroscopic particles • Interaction between particles- repulsive contact forces • Why are they studied? • Use • Properties and behavior

3. Indeterminants • The stacking of cannon balls • Hyper-static Equilibrium ( 6 Contact Points ) • Stable Equilibrium ( Any 3 Contact Points ) • Contacts become random

4. Hysteresis • For a particle at rest on multiple surfaces, direction of frictional force can’t be determined • Without prior knowledge of system forces can be determined

5. Statistics • Indeterminacies make straight forward analytical approaches difficult • Numerous grains in material furthers this difficulty • Statistical methods are a natural way to analyze this type of system

6. Probability Distribution • Distributions can be used to study general properties of forces in the system • Systems undergoing different processes can be identified

7. Most likely shear Ft is about its mean value. All other forces most probable value is near its mean. • Both compression forces share similar probability at high forces but shear Ft are more likely to be bigger then Fn

8. Radial Distributions • Can be used to study the direction of propagating forces • Net forces on system and propagation of forces can be extrapolated.

9. 12 contact points represents the 6 equilibrium points of the two configurations • Represents two different configurations

10. Correlation • Finds the linear dependence of forces between two grains as a function of separation • If defined as where F(x) is the sum of contact forces on a grain at x • Can be use to find force chain lengths

11. Shear system has longer probable chains lengths in y direction then x • Compression has equally probability in both directions

12. Clusters Connected and occupied sites

13. Percolation Theory • What is Percolation theory? • Numbers and properties of the clusters • f= force • fc= critical threshold force • Elaborate later on scaling exponents and function • Use of the Percolation theory model • s= random grain size; fc= critical threshold;  and  are scaling exponents; =scaling function

14. Mean Cluster Size • S is the mean cluster size • ns(p) is the number of clusters per lattice site • A general form of the moment • N=system size i.e. number of contacts in the packing

15. Why Percolation Theory? • Probability of connectivity • f=0, f=1 • f< fc, f> fc • Force network inhomogeneity in granular materials • Quantification of force chains • Threshold, fc, small and large f • Force network variation- statistical approach • Around fc, the system shows scale invariance • Power-law behavior of our scaling exponents and scaling function • Suggests systems with this behavior have same properties

16. N-Φm2 and (f-fc)N1/2v are rescale with B and A respectively • Φ = 0.89 ± 0.01 , v = 1.6 ± 0.1 • A and B are a function of polydispersity, pressure and coefficient of friction

17. Plot show similar features • Problem in calculating fc • For proper scaling in x-axis proper centering is needed